Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

Parametrizations of elliptic curves by Shimura curves and by classical modular curves

Fix an isogeny class

Cellular mechanisms for resolving phase ambiguity in the owl's inferior colliculus

Both mammals and birds use the interaural time difference (ITD) for localization of sound in the horizontal plane. They may localize either real or phantom sound sources, when the signal consists of a narrow frequency band. This ambiguity does not occur with broadband signals. A plot of impulse rates or amplitude of excitatory postsynaptic potentials against ITDs (ITD curve) consists of peaks and troughs. In the external nucleus (ICX) of the owl's inferior colliculus, ITD curves show multiple peaks when the signal is narrow-band, such as tones. Of these peaks, one occurs at ITDi, which is independent of frequency, and others at ITDi ± T, where T is the tonal period. The ITD curve of the same neuron shows a large peak (main peak) at ITDi and no or small peaks (side peaks) at ITDi ± T, when the signal is broadband. ITD curves for postsynaptic potentials indicate that ICX neurons integrate the results of binaural cross-correlation in different frequency bands. However, the difference between the main and side peaks is small. ICX neurons further enhance this difference in the process of converting membrane potentials to impulse rates. Inhibition also appears to augment the difference between the main and side peaks.

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

A joint hazard and time scaling model to compare survival curves.

To provide a more general method for comparing survival experience, we propose a model that independently scales both hazard and time dimensions. To test the curve shape similarity of two time-dependent hazards, h1(t) and h2(t), we apply the proposed hazard relationship, h12(tKt)/ h1(t) = Kh, to h1. This relationship doubly scales h1 by the constant hazard and time scale factors, Kh and Kt, producing a transformed hazard, h12, with the same underlying curve shape as h1. We optimize the match of h12 to h2 by adjusting Kh and Kt. The corresponding survival relationship S12(tKt) = [S1(t)]KtKh transforms S1 into a new curve S12 of the same underlying shape that can be matched to the original S2. We apply this model to the curves for regional and local breast cancer contained in the National Cancer Institute's End Results Registry (1950-1973). Scaling the original regional curves, h1 and S1 with Kt = 1.769 and Kh = 0.263 produces transformed curves h12 and S12 that display congruence with the respective local curves, h2 and S2. This similarity of curve shapes suggests the application of the more complete curve shapes for regional disease as templates to predict the long-term survival pattern for local disease. By extension, this similarity raises the possibility of scaling early data for clinical trial curves according to templates of registry or previous trial curves, projecting long-term outcomes and reducing costs. The proposed model includes as special cases the widely used proportional hazards (Kt = 1) and accelerated life (KtKh = 1) models.